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Fixed points and stability of an integral equation: nonuniqueness. (English) Zbl 1066.45002
Summary: We consider a paper of J. Banaś and B. Rzepka [ibid. 16, No. 1, 1–6 (2003; Zbl 1015.47034)] which deals with existence and asymptotic stability of an integral equation by means of fixed-point theory and measures of noncompactness. By choosing a different fixed-point theorem, we show that the measures of noncompactness can be avoided and the existence and stability can be proved under weaker conditions. Moreover, we show that this is actually a problem about a bound on the behavior of a nonunique solution. In fact, without nonuniqueness, the theorems of stability are vacuous.
MSC:
45G10Nonsingular nonlinear integral equations
47H09Mappings defined by “shrinking” properties
47N20Applications of operator theory to differential and integral equations
45M05Asymptotic theory of integral equations
45M10Stability theory of integral equations